Coulomb's Law

Coulomb's law is the electrostatic analogue of Newton's gravitation — it gives the force between two stationary point charges. But unlike gravity, charges come in two signs, the constant is enormous, and one needs to be careful with vectors and media.

Concept

Two point charges $q_1$ and $q_2$ separated by distance $r$ in vacuum exert on each other a force $F = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2},$ along the line joining them. The proportionality constant is $k = \frac{1}{4\pi\varepsilon_0} \approx 9.0\times 10^9\,N\cdot m^2/C^2,$ where $\varepsilon_0 = 8.854 \times 10^{-12}\,C^2/(N\cdot m^2)$ is the permittivity of free space.

Vector form. Let $\hat r_{12}$ be the unit vector from charge $1$ to charge $2$. The force on charge $2$ due to charge $1$ is $\vec F_{12} = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}\,\hat r_{12}.$ Newton's third law gives $\vec F_{21} = -\vec F_{12}$. The sign of the product $q_1 q_2$ determines whether the force is repulsive ($+$) or attractive ($-$).

Force in a medium. Replace $\varepsilon_0$ by $\varepsilon = K\varepsilon_0$ where $K$ is the dielectric constant of the medium: $F_\text{medium} = \frac{F_\text{vacuum}}{K}.$

Superposition. With many charges, the net force on charge $q_0$ is the vector sum of pairwise forces: $\vec F = \sum_{i=1}^{n} \vec F_{0i}.$

Derivation

The inverse-square form was found experimentally with a torsion balance (Coulomb, 1785). It is consistent with Gauss's law in three dimensions. Consider two charges at positions $\vec r_1$ and $\vec r_2$. Define $\vec r = \vec r_2 - \vec r_1, \quad r = |\vec r|, \quad \hat r = \vec r/r.$ Then the force on the second charge is $\vec F_{12} = \frac{kq_1 q_2}{r^2}\hat r = \frac{kq_1 q_2}{r^3}\vec r.$ The $r^3$ form is convenient because $\vec r$ already contains the direction.

For three charges placed on a line at positions $0, a, 2a$, the force on the middle one (charge $q_2 = q$) due to outer charges $q_1 = q_3 = q$ cancels by symmetry. If $q_3 = -q$, the forces add to push $q_2$ towards $q_3$.

Worked Example

Two point charges $q_1 = +3\,\mu C$ and $q_2 = +5\,\mu C$ sit $20\,cm$ apart in air. Find the force on $q_2$.

$F = \frac{(9\times10^9)(3\times10^{-6})(5\times10^{-6})}{(0.20)^2} = \frac{0.135}{0.04} = 3.375\,N.$ The force is repulsive, directed away from $q_1$.

If the same configuration is immersed in water ($K = 80$), the force becomes $F' = 3.375/80 \approx 0.042\,N.$

Common Confusions

• Coulomb's law is exact only for point charges. For extended objects you must integrate.
• $F \propto 1/r^2$ assumes the inverse-square structure of 3D space. In other dimensions this would change.
• Signs matter. Use absolute values for magnitude and reintroduce direction with a unit vector — don't double-count.
• Force in medium uses $K$, not just $\varepsilon_0$. Many students forget that water massively reduces electrostatic forces.

Key Takeaways

• $F = k q_1 q_2 / r^2$, repulsive for like charges, attractive for unlike.
• Vector form: $\vec F_{12} = k q_1 q_2 \hat r_{12}/r^2$.
• Force in a medium: divide vacuum value by dielectric constant $K$.
• Multiple charges: use vector superposition.